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Mirrors > Home > ILE Home > Th. List > nfald | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎ𝑦𝜑 |
nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfald | ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1512 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | nfald.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 2, 3 | alrimih 1462 | . 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓) |
5 | nfnf1 1537 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓 | |
6 | 5 | nfal 1569 | . . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓 |
7 | hba1 1533 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦∀𝑦Ⅎ𝑥𝜓) | |
8 | sp 1504 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓) | |
9 | 8 | nfrd 1513 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓)) |
10 | 7, 9 | hbald 1484 | . . 3 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) |
11 | 6, 10 | nfd 1516 | . 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥∀𝑦𝜓) |
12 | 4, 11 | syl 14 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 Ⅎwnf 1453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-4 1503 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: dvelimALT 2003 dvelimfv 2004 nfeudv 2034 nfeqd 2327 nfraldw 2502 nfraldxy 2503 nfiotadw 5163 nfixpxy 6695 bdsepnft 13922 |
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